Is the origin included in polar half-lines/radial lines?

Question

In the complex plane, $\arg(z)=\alpha$ defines a half-line starting at the origin at an angle $\alpha$ from the positive real axis, however the origin itself is not included in the half-line as $\arg(0)$ is undefined. In the polar coordinate system, $\theta=\alpha$ defines a half-line with $\alpha$ being the angle from the initial line. Is the origin included in this polar half-line? As $x=r\cos(\theta)$ and $y=r\sin(\theta)$, $r$ is allowed to be $0$ so I don't think we have the same kind of restriction as the complex plane. I do understand that $\theta$ is not unique for $r=0$ but in this case, that should not matter for half-lines, should it? As in, we can just allow all half-lines from the origin to include the origin.